Abstract: Inflationary cosmology attempts to provide a natural explanation for the flatness and homogeneity of the observable universe. In the context of reversible (unitary) evolution, this goal is difficult to satisfy, as Liouville’s theorem implies that no dynamical process can evolve a large number of initial states into a small number of final states. We use the invariant measure on solutions to Einstein’s equation to quantify the problems of cosmological fine-tuning. The most natural interpretation of the measure is the flatness problem does not exist; almost all Robertson-Walker cosmologies are spatially flat. The homogeneity of the early universe, however, does represent a substantial fine-tuning; the horizon problem is real. When perturbations are taken into account, inflation only occurs in a negligibly small fraction of cosmological histories, less than 10-6.6×10^7. We argue that while inflation does not affect the number of initial conditions that evolve into a late universe like our own, it nevertheless provides an appealing target for true theories of initial conditions, by allowing for small patches of space with sub-Planckian curvature to grow into reasonable universes.

In English: our universe looks very unusual. You might think we have nothing to compare it to, but that’s not quite right; given the particles that make up the universe (or the quantum degrees of freedom, to be technical about it), we can compare their actual configuration to all the possible configurations they could have been in. The answer is, our observed universe is highly non-generic, and in the past it was even more non-generic, or “finely tuned.” One way of describing this state of affairs is to say that the early universe had a very low entropy. We don’t know why; that’s an important puzzle, worth writing books about.

Part of the motivation of this paper was to put some quantitative meat on some ideas I discussed in my book. The basic argument is an old one, going back to Roger Penrose in the late 1970’s. The advent of inflation in the early 1980’s seemed to change things — it showed how to get a universe just like ours starting from a tiny region of space dominated by “false vacuum energy.” But a more careful analysis shows that inflation doesn’t really change the underlying problem — sure, you can get our universe if you start in the right state, but that state is even more finely-tuned than the conventional Big Bang beginning.

We revisit this question, bringing to bear some mathematical heavy machinery developed in the 1980’s by Gary Gibbons, Stephen Hawking, and John Stewart. Previous discussions have invoked general ideas of entropy or reversibility, but we were able to do a relatively down-to-earth calculation using conventional cosmological models. And we tried our best to explicitly list all of the caveats of the argument, which is important in a context like this where we don’t know all the rules.

We find that inflation is very unlikely, in the sense that a negligibly small fraction of possible universes experience a period of inflation. On the other hand, our universe is unlikely, by exactly the same criterion. So the observable universe didn’t “just happen”; it is either picked out by some general principle, perhaps something to do with the wave function of the universe, or it’s generated dynamically by some process within a larger multiverse. And inflation might end up playing a crucial role in the story. We don’t know yet, but it’s important to lay out the options to help us find our way.

Hmmm…it seems like your paper can also conclude that perhaps God fine-tuned the universe to be what it is so that we can exist. Good work. Well, you can also explain the findings by appealing to a “multiverse”. However, both the multiverse and God are unobservable. So let’s just choose God.

http://twistedphysics.typepad.com Jennifer Ouellette

I choose the multiverse. It’s far less cranky and vindictive.

http://www.sonic.net/~rknop/blog Rob Knop

Does something like the ergodic hypothesis apply? Even if the conditions that lead to a false vacuum and inflation are very rare, the ergodic hypothesis says that eventually they will happen by chance (for suitable definitions of “eventually” when it’s not clear exactly what we mean by time).

And, yeah, that we’re in this unlikely universe means I’m implicitly relying on something like a weak anthropic argument– but weak anthropic arguments ultimately are just selection effects.

http://sacrilicio.us Matunos

Haven’t you heard? The proton is smaller than we thought. All bets are off.

Humbert Hobart

Physical systems are not reversible.

The universe is inhomogeneous (unbounded discrete fractal).

All physical events are causal.

Arguments based on “multiverses”, “anthropic reasoning” and an acuasal “beginning” of nature are bad natural philosophy, foisted upon us poor souls by deluded Platonist glass-bead game players.

Big Boffin

PROTON RADIUS

High precision empirical measurement = 0.84 fermi

QED-based estimates + 0.877 to 0.9 fermi

Discrete Scale Relativity = 0.814 fermi

GOOOOAAAALLL! Notify the octopus!

SCORE: DSR 1/QED 0

TJ

I believe you’re misinformed. Your statements about the universe are in disagreement with the Cosmological Principle. And Special Relativity clearly indicates that there be causally separated events.

I suppose you may disagree with these theories, but there is a great deal of experimental evidence to support them.

http://feralboy12.com feralboy12

I don’t know what you’re talking about either, but my theory involves something I call the Plywood Principle.
P.S. I am not a crank.

http://www.astro.multivax.de:8000/helbig/helbig.html Phillip Helbig

I don’t have the reference handy now, but can dig it up if necessary. Back in the 1990s, John D. Barrow wrote a paper in the Physical Review D (certainly a respectable journal) in which he concluded “there is no horizon problem for inflation to solve” (perhaps not an exact quote, but if not, then rather close). I remember reading the paper, and briefly discussing it with Barrow at the Texas Symposium in Munich in 1994, but haven’t seen it cited much.

Barrow is not a crackpot (though he did accept some Templeton money), PhysRevD is a respectable journal and this is an important problem in cosmology. Thus, I would expect the paper to be highly cited, unless another paper has refuted it. Since the first apparently hasn’t happened, have I missed the second?

In any case, I’d be interested in Sean’s (and any other informed) opinion.

http://telescoper.wordpress.com Peter Coles

I remember having a lot of discussions with George Ellis way back in the 90s about this issue. I strongly agree that what inflation does is merely to push the fine-tuning problems back to an earlier epoch where they are effectively under the carpet (or beyond the horizon, if you prefer a different metaphor). In fact we were planning to write a sort of spoof of Galileo’s “Dialogue concerning the Two Chief World Systems” featuring characters with names like “Inflatio” and “Anthropicus” …. but never got around to it.

http://telescoper.wordpress.com Peter Coles

PS. To anyone of a Bayesian persuasion statements about probability are statements about the extent to which a logical proposition or theory is credible given the observations. Sean is saying that the observed Universe is improbable given our theory of inflation. The question whether our observed Universe renders inflationary theory improbable is not the same, and is indeed much more interesting…

http://www.astro.multivax.de:8000/helbig/helbig.html Phillip Helbig

In your book with George, I think you make this point rather clearly with respect to the flatness problem. Can the horizon problem be solved in the same way? I can see the solution to the first (and am surprised that many even still consider it a problem) but not the second (at least not from the arguments in your book with George).

We consider the optimal positioning of an even number of identical crew members in a coxless racing boat so as to avoid the presence of a sideways wiggle as the boat is propelled forward through the water. We show that the traditional (alternate port and starboard) positioning always possesses an oscillating nonzero transverse moment and associated wiggling motion and that the problem of finding the zero-moment positions is related to a special case of the subset sum problem. We find the one (known) zero-moment rig for a racing Four and show that there are four possible such rigs for a racing Eight, of which only two are known. We show that only balanced boats with crew numbers that are divisible by four can have the zero-moment property and give the 29 zero-moment solutions for racing Twelves, which have zero transverse moments. Some aspects of unbalanced boats in which the numbers of port and starboard oars are unequal are also discussed.

I knew that moving to Cambridge would bring John over to the dark side.

reminding me why I miss the QJRAS. This and similar articles by Longair, Rees, George Ellis, Harrison etc were (and are) some of my favourite reading.

http://www.astro.multivax.de:8000/helbig/helbig.html Phillip Helbig

“The question whether our observed Universe renders inflationary theory improbable is not the same, and is indeed much more interesting…”

The probability of the model given the data vs. the probability of the data given the model: the distinction can be quite important. This example illustrates the point: My data is that a person is pregnant, my model (or theory) is that this person is female. The probability of the data, given the model (and no other information) is about 3%. The probability of the model, given the data, is 100%.

Here’s the Barrow reference: http://adsabs.harvard.edu/abs/1995PhRvD..51.3113B (It’s from 1995, but the Texas Symposium was in 1994; I suppose I had read a preprint originally.) Here’s the quote (from the abstract; see text for details): It is shown that homogeneous cosmologies display no “isotropy problem” for inflation or quantum cosmology to solve.. OK, “isotropy problem” and not “horizon problem”, but that is just a naming issue. (We observe isotropy and simple arguments make it difficult to understand since in the past the horizon scale was smaller than areas of, say, similar CMB temperature on the sky today.)

Instead of updating the Dialogue, what about something based on the second Blackadder series? Let’s see, Peter Coles would be Blackadder, I would be Baldrick, Sean would be Melchett and Virginia Trimble would be Elizabeth. And, of course, Rocky Kolb would be Lord Flashheart.

http://blueollie.wordpress.com ollie

So, is this like saying that “the set of universes that are similar to ours” forms a set of small measure, or is it saying that it forms a “non dense” set (in an appropriate topology)?

Paul

“I choose the multiverse. It’s far less cranky and vindictive.”

God loves you, Jennifer, the multiverse doesn’t. You’ll see after death, as we all shall see.

http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

brandon — there’s a link right there in the post.

ollie– it’s a set of small measure; that’s where the number quoted in the abstract comes from.

http://blueollie.wordpress.com ollie

Thanks Sean.

DaveH

So the observable universe didn’t “just happen”

What does this mean, in the context? That inflation + the anthropic principle isn’t sufficient?

Elliot Tarabour

I believe that we need to consider some form of natural selection occurring within the multiverse. I have elsewhere suggested (over at NPR 13.7 blog) that we find ourselves in a large, inflated, complex universe, because their is selective survival advantage to baby universes which favor complex information processing. The idea of cosmic natural selection is not new. Lee Smolin proposed it a number of years ago. He however thinks that black hole production confers selective advantage. I think it is the generation of complex information.

e.

Brian Mingus

Sean et al,

I find this paper relevant with respect to another recent result, that being the purported new mass of the proton, and fine tuning in general.

Consider the conundrum that we are faced with: namely, particle accelerators have replicated this result for decades and it may be wrong. Thus, how can we trust just one particle accelerator producing the new result? The utility of repeatability in science seems to be thrown out the door.

Now suppose that the actual mass of the proton is off by another n%, but that detecting it requires an even more sensitive experiment. This means we will adjust the standard model to fit the most recent result, but given the precision of the model we might be wrong.

Now suppose that the universe is extremely fine tuned, and further suppose that in order to get to a “theory of everything” you need to be able to measure the precision of x to n digits. At this point there are three forks in the road:

1) the standard model at time t is consistent with all available evidence thus produced and is correct
2) the standard model at time t is consistent with all available evidence and is wrong, but will be corrected in the future when a more sensitive experiment is designed
3) the standard model at time t is consistent with all available evidence and is wrong and no experiment can ever make a precise enough measurement (at least vaguely akin to a chaotic system). You might even be able to prove that you can’t make the measurement.

Is this overgeneralization of “fine tuning”, or do you consider it plausible that we could find ourselves in position 3?

Big Boffin

Is ANY part of the multiverse hypothesis scientifically testable?

Perhaps what the little circle of glass-bead game players need is for M. Kaku to step into the center and “read the mind of God” for the group?

Untestable postmodern pseudoscience. Ain’t it unreal, man!

Felix

So Sean puts some precision into the ‘fine tuning’ of the universe. Good work.

God and multiverse are the same thing. If God is part of this universe, then it is not God right? If it is outside, and did its ‘creation’, then by definition it is on another ‘universe’ – thus multiverse. Fine tuning automatically imply the existence of ‘something’ outside of this universe before its existence. It also means it is impossible for any intelligence in this universe to know what’s ‘outside’, just that the ‘outside’ exists.

A nonlinear dynamical system is deterministic and fully causal, and
yet not entirely predictable. It can go from quasi-classical behavior,
including periodic behavior, into full or partial chaotic behavior,
and back again.

A NLDS has a definite “arrow” and you can call it the arrow of time,
or the arrow of determinism, or the arrow of causality. They are all
different “facets of the same crystal”.

The key issue here is that you do not have to invent untestable
hypothetical “multiverse” pipe-dreams in order to explain the arrow.
If you have an NLDS on any scale, microscopic or macroscopic, then you
have a local arrow for that system on that scale.
This is the reason you cannot unscramble your scrambled eggs. It has
nothing remotely to do with the Big Bang, or pre-Big Bang physics.

Then the question is: how common are NLDS? My intuition and
observations suggest that the answer is: highly ubiquitous.

I would ask: what well-studied, and observed at high resolution,
physical systems are not NLDS?

joel rice

This fine-tuning stuff might not be needed – along with God, and Natural Selection in Physics, and the Anthropic Principle, and Multiverses. Suppose merely that the Algebraic Design of the world requires the construction of atoms – the whole collection of atomic structures. That forbids any kind of crummy tuning that would derail construction. You might not even need any “constants” . In fact, the structure would force the constants to be what they have to be for construction to be possible. Evidently Matter is just as much an issue of construction as Space being a collection of all its subspaces, in a multi-vector. Every atomic structure imposes constraints on the properties of forces. You might imagine fiddling with the properties of Hydrogen, but if you must account for Helium, then less fiddling is allowed if you expect consistency across all atoms. That is why I think it is a huge oversight for Physics to ignore Octonion algebra – it requires associations of building blocks – because it is nonAssociative. We are surrounded by and are made of associations of elementary things. Every atom and every molecule is an association.

But why not seek Geometric Principles for the unified modeling of an eternal multi-scaled cosmos, with one reasonably limited set of fundamental constants that apply on all scales?

It’s not that I’m against algebra, it’s just that it tends to be an approximation for underlying geometric fundamentals.

Discrete Scale Relativity offers a new paradigm of this type. It has passed 39 fundamental retrodictions and makes at least 10 definitive predictions. It’s prediction for the radius of the proton is closer to the brand new high precision measurement than anything the standard hep model has barfed up with much straining. Discrete Scale Relativity does it with a simple calculation using the Kerr-Newman metric and the correct G value for the interiors of atomic scale systems.

joel rice

Boffin – multivector algebra ( ie, Clifford Algebra ) already is as geometrical as it gets , as well as being as fundamental as it gets, although if we generate Clifford algebras by picking the number of vectors and the signature, it seems to depend on the heavy human hand. Luckily Complex Quaternions can go like 1 + x + y + z + xy +yz + zx + xyz for plain old three dimensional spatial structure. It is also the same thing as Pauli algebra when expressed as 2×2 complex matrices. Talk about Dual Use Technology ! Besides, it is the even subalgebra of complex octonions, which have -+++ and +— signatures, and no others, which looks like it gets Minkowski spacetime automatically, and can not be otherwise.
So I do not see algebra being an approximation for geometrical fundamentals, and find the whole idea of getting rid of these infernal ‘constants’ an endlessly facinating possibility.

James

Sean, I’m not quite sure I understand the nature of this question – to what extent does the cosmology community already recognise that the flatness problem is not really a problem? Has this paper just put some meat on the bones of an already-understood argument, or has the point not been properly made before (I’m afraid I don’t have access to those prior papers you cite).

http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

I think the consensus in the cosmology community is that the flatness problem is real. Hawking and Page previously pointed out that the canonical measure suggests otherwise, but they didn’t really push it, and it certainly didn’t become conventional wisdom.

SteveB

I do not see why a small probability for a state out of a near infinity of possible states surprises anyone. I do see value in comparing possible states as a way to uncover some fundamental operating phenomena; I just don’t see that “our universe looks very unusual.”

For example, I have not been hit by a meteor weighing 1.0 kg travelling 100 m/s.
I have also not been hit by a meteor weighing 1.01 kg travelling 100 m/s.
I have also not been hit by a meteor weighing 1.001 kg travelling 100 m/s.
…
I have also not been hit by a meteor weighing 1.0 kg travelling 100.1 m/s.
…
rotating meteors, iron vs. carbonaceous meteors, “just missing me” meteors and all the x,y values for that.
…
Wow there are a lot of ways for a meteor to hit or almost hit me! Since no meteor has yet done that, I must be very, very special indeed. ….Not! Actually, I am not surprised at all that a meteor has not startled me; it is not unusual.

I would solve the meteor problem by experimentally measuring the number of meteors striking the earth in a year with a measured size distribution, a little planetary astronomy to look at possible variations to that average number, and then calculating the probability a large enough object meets my criterion in some period of time, and most importantly — adding the error bars which are computed from each and every measurement and assumption I use.

Kevin– It’s an interesting idea, and Verlinde is certainly a smart and creative guy. It might turn out to be important, or to be relatively trivial. I haven’t written about it because I haven’t had the time to really digest the paper.

http://magicdragon.com Jonathan Vos Post

re #21: I’ve argued since the very month that Smolin put forth the idea of Cosmic Natural Selection, that Population Genetics shows advantage to sexually reproducing species, so we should deduce a multiverse cosmology where pairs of universes pass physical constants on to baby universes. Then the Genetic Algorithm kicks in, per the breakthrough book Adaptation in Natural and Artificial Systems, John H. Holland, 1975, (republished by The MIT Press, 1992), which I beta-tested 1974 for prof. Holland in preprint while I was in grad school.

karaktur

Mr. Mayer might say,

Entopy
Is workin’
Against me

And entropy
Wants to
Bring
Me
Down

joel rice

In case someone thinks I was blowing smoke above, see the last page of Robert Hermann’s
“Spinors, Clifford and Cayley Algebras” ( Math Sci Press 1974 page 272). well, he gets half
of it.
Sean : considering that the argument I made above might blow a hole in your position, just wondering if you care to comment.

Shantanu

Sean and other inflation afficianados, what do you people think about this which talks about torsion as an alternative to
regular scalar field inflation?

Hmmm…it seems like your paper can also conclude that perhaps God fine-tuned the universe to be what it is so that we can exist. Good work. Well, you can also explain the findings by appealing to a “multiverse”. However, both the multiverse and God are unobservable. So let’s just choose God.

Well, if it was God doing the fine-tuning then He did a shoddy job of it. Shoddy enough that He does not deserve to be worshiped for it. Unless our existence was not the point. In which case God should either be prosecuted for crimes against humanity, or sued for negligence. Much, much preferable, theologically, methinks, to choose a mindless multiverse.

bittergradstudent

@Shantanu #40

I don’t know if I’m an ‘inflation aficionado’, but torsion is a really heavy handed way to fix the issues solved by inflation. In particular, in the presence of torsion, test particles no longer follow the same geodesics, and you get new spin-orbit coupling effects. If you’re going to add torsion to your theory, you have to be extremely careful about making sure that solar system tests of relativity aren’t violated. Not to mention things like the Hulse Pulsar, which looks pretty consistent with Einstein gravity, at least in the evolution of its orbital period.

The paper claims that the magnitude of the torsion tensor is small, so maybe this isn’t violated. But such things require care, and I would be skeptical about torsion being the answer until the issue of consistency with existing observations was settled.

Shantanu

bittergradstudent,
In hulse-taylor binary pulsar you have 3 unknowns and 4 observables, which is how you
can test GR. In an alternate theory of gravity, you will have one more unknown and I don’t
see how you can test torsion theories with binary pulsar. Also I am not sure PPN formalism
deals with torsion.

bittergradstudent

Shantanu,

You’d have to actually do a calculation that shows me that the radiation reaction is the same. The Hulse-Taylor Pulsar has tiny, tiny error bars–if you change the radiation reaction one bit, you’re going to change the shape of that graph, and torsion changes everything. Same with the solar system tests of GR.

If you’re going to make me believe in torsion, you’re going to have to show me that it isn’t already disproven.

bittergradstudent

(sorry about the double post)

Or, put another way, an inflaton adds one degree of freedom to your theory, and an unknown potential function. Perhaps less than that if you can make it an emergent phenomenon of particle physics.

In D=4, a torsion tensor adds 24 independent tensor components (6 antisymmetric components to a 10×10, times four coordintes) to your theory, complete with unknown dynamics to generate those components. It’s a much less sparse explanation. Absent a reason other than not liking inflation, I don’t see much of a reason to want torsion over inflation. Of course, every theory should be explored and explained, but I’m unconvinced, and am skeptical that its even a possibility unless I can be shown that a torsion theory passes astrophysical tests with a large enough torsion tensor to matter cosmologically intact. And of course you could deal with torsion in a PPN formalism–torsion just modifies the Christoffel symbols, and, in the end, PPN is really an expansion of the Christoffel symbols.

Shantanu

bittergradstudent,
the radiation reaction rate depends upon the masses of the binary pulsar, distance,inclination etc. All of these are determined using the post-keplerian formalism developed
by Damour and Durelle. However since the number of observables are more than the number of unknown, you can test self-consistency of the theory as system is over-determined. whereas with the inclusion of torsion, no of unknowns will be = no of equations and I don’t think you can test anything.

bittergradstudent

Even if you can’t get precise measurements of the parameter, you can get bounds on them. I wouldn’t be surprised if torsion made it so that stable bound orbits didn’t exist in the strong field case, for example. The shape of the radiation curve might change. It will also either violate gauge invariance or the equivalence principle, depending on whether the new Maxwell tensor is $latex nabla_{[a}A_{b]}$ or $latex nabla_{[a}A_{b]} -T^{c}{}_{ab}A_{c}$ . The first one is no longer gauge invariant, while the second one is gauge invariant, but will treat the Maxwell field differently than ordinary matter and violate the equivalence principle.

There are a lot of unobserved effects that torsion predicts. I’m pretty sure that there is a strenuous lower bound on it. I just don’t know how low it is.

http://opines.mythusmage.org Alan Kellogg

What if conditions were such in the very early universe that mass was impossible. No mass, no speed limit. I’ve got further thinking along this line, but that would take up space. BTW, I have no idea if I am a crank, because cranks are often unaware of their crankhood.

Shantanu

bittergradstudent, what do you think of this recent paper by BJ which discusses the role of torsion
in cosmology here am surprised there is no blog
discussing this paper.

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Cosmic Variance

Random samplings from a universe of ideas.

About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson.
Here are some of his favorite blog posts, home page, and email: carroll [at] cosmicvariance.com .